Question

Verify that each of the following force fields is conservative. Then find, for each, a scalar potential \(\phi\) such that \(\mathbf{F}=-\mathbf{\nabla} \phi\).\(\mathrm{F}=y \sin 2 x \mathbf{i}+\sin ^{2} x \mathrm{j}\)

Short answer

The field is conservative, and \(\phi(x, y) = \frac{y}{2} \cos 2x + C\).

Step-by-step solution

Check if the Force Field is Conservative

A force field \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j}\) is conservative if the condition \(\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}\) holds. For \(\mathrm{F} = y \sin 2x \mathbf{i} + \sin^{2} x \mathbf{j}\), identify \(P = y \sin 2x\) and \(Q = \sin^{2} x\).

Compute Partial Derivatives

Calculate \(\frac{\partial P}{\partial y}\) and \(\frac{\partial Q}{\partial x}\): \[\frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(y \sin 2x) = \sin 2x\] \[\frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(\sin^{2} x) = 2 \sin x \cos x = \sin 2x\]

Verify the Conservative Condition

Compare \(\frac{\partial P}{\partial y}\) and \(\frac{\partial Q}{\partial x}\). Since both are equal to \(\sin 2x\), the force field is conservative.

Introduce Scalar Potential \(\phi\)

Since \(\mathbf{F} = -\mathbf{abla} \phi\), we can write \(\phi(x, y)\) such that \(\frac{\partial \phi}{\partial x} = -P = -y \sin 2x\) and \(\frac{\partial \phi}{\partial y} = -Q = -\sin^{2} x\).

Integrate with Respect to \(x\)

To find \(\phi\), integrate \(\frac{\partial \phi}{\partial x} = -y \sin 2x\) with respect to \(x\): \[\phi(x, y) = -y \int \sin 2x \ d x = -y \left(-\frac{1}{2} \cos 2x\right) + g(y) = \frac{y}{2} \cos 2x + g(y)\] where \(g(y)\) is a function of \(y\) only.

Determine \(g(y)\)

Differentiate \(\phi(x, y)\) with respect to \(y\) and equate it to \(\frac{\partial \phi}{\partial y} = -\sin^2 x\), then solve for \(g(y)\). \[\frac{\partial \phi}{\partial y} = \frac{1}{2} \cos 2x + g'(y) = -\sin^2 x\] Since \[\frac{1}{2} \cos 2x = \cos^2 x - \sin^2 x = \cos^2 x - (1 - \cos^2 x) = 2 \cos^2 x - 1\] \ This ensures: \[\sin^2 x = -g'(y)\] Thus, \[g(y) = C\] where \(C\) is a constant.

Key concepts

Partial Derivatives

Partial derivatives are a fundamental tool in calculus used to analyze functions of multiple variables. When dealing with functions like our force field \(\mathbf{F}\), which depend on variables (x and y), understanding partial derivatives is crucial.

Let's recall the partial derivative simply measures how a function changes as one of its variables is adjusted. For our force field, we evaluate \(\frac{\partial P}{\partial y}\) and \(\frac{\partial Q}{\partial x}\). Here's how it applies:

\[\frac{\partial P}{\partial y} = \sin 2x,\] translating to checking how P (\(\mathbf{P} = y \sin 2x\)) changes as y varies.

\[\frac{\partial Q}{\partial x} = \sin 2x,\] translating to checking how Q (\(\mathbf{Q} = \sin^{2} x\)) changes as x varies.

If these derivatives hold equal, the force field remains conservative. For our example, both are equal, so our field is confirmed conservative.

Scalar Potential

Once we know a force field is conservative, the next step is to find a scalar potential, usually denoted as \(\phi\). This potential function helps describe the force field more compactly.

We set out to find \(\phi\) such that \(\mathbf{F} = -\abla \phi\). This means:

• \[\frac{\partial \phi}{\partial x} = -P = -y \sin 2x,\]
• \[\frac{\partial \phi}{\partial y} = -Q = -\sin^{2} x.\]

We then integrate with respect to x:

\[\phi(x, y) = -y \int \sin 2x \ dx = \frac{y}{2} \cos 2x + g(y),\]

where \(\int\) represents the integral and g(y) is a function of y. To determine g(y), differentiate \(\phi(x, y)\) with respect to y and resolve for g(y). By comparing, we find:

\[g(y) = C\]

where C is a constant. Hence, we achieve the scalar potential.

Vector Calculus

Vector calculus is essential when exploring fields in physics and engineering. It allows us to work with vector quantities and helps us analyze things like force fields.

Key aspects of vector calculus include:

• Gradient (\( \abla \phi \)): This operation, when applied to a scalar potential \(\phi\), yields a vector field. For our case, \(\mathbf{F} = -\abla \phi \).
• Divergence (\( \abla \cdot \mathbf{F} \)): Measures how much a vector field spreads out from a point. A zero divergence often indicates potential conservativeness.
• Curl (\( \abla \times \mathbf{F} \)): Measures the rotation of the field around a point. A zero curl suggests the force field might be conservative.

Understanding these operations allows you to assess and find properties of vector fields, making vector calculus an invaluable area of study.